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A note on 3-D simple points and simple-equivalence
Gilles Bertrand, Michel Couprie, Nicolas Passat
To cite this version:
Gilles Bertrand, Michel Couprie, Nicolas Passat. A note on 3-D simple points and simple-equivalence.
Information Processing Letters, Elsevier, 2009, 109 (13), pp.700-704. �10.1016/j.ipl.2009.03.002�. �hal-
00622395�
A note on 3-D simple points and simple-equivalence
Gilles Bertrand
(a), Michel Couprie
(a)and Nicolas Passat
(b)(a) Universit´e Paris-Est, Laboratoire d’Informatique Gaspard-Monge, ´Equipe A3SI, ESIEE Paris, France (b) LSIIT, UMR 7005 CNRS/UdS, Strasbourg University, France
e-mail: (g.bertrand,m.couprie)@esiee.fr, passat@unistra.fr
Keywords: Digital topology, cubical complexes, simple point, fundamental group.
1. Introduction
Topology-preserving operators, like homotopic skeletonization, are used to transform the geometry of an object while leaving unchanged its topological characteristics. In discrete grids (Z2 or Z3), such a transformation can be defined thanks to the notion of simple point: intuitively, a point of an object is called simple if it can be deleted from this object without altering topology. This notion, pionneered by Duda, Hart, Munson, Golay and Rosenfeld [12], has since been the subject of an abundant literature. In particu- lar, local characterizations of simple points have been proposed, which enable efficient implementations of thinning procedures.
In [11], the authors study some configurations where an object X strictly contains an object Y , topo- logically equivalent to X , and where X has no simple point. We call such a configuration a lump (an exam- ple of lump is given in Fig. 1a). Lumps cannot appear inZ2, but they are not uncommon inZ3, where they may “block” some homotopic thinning procedures and prevent to obtain globally minimal skeletons. The ex- istence of certain such objects, like the one of Fig. 1a, illustrates a “counter-property” of simple points: delet- ing a simple point always preserves topology, but one can sometimes delete a non-simple point while preserv- ing topology. In Fig. 1a, the point x is not simple (in Sec. 4 we give a definition and a characterization which enable the reader to verify this claim), but we can see that if we remove x, intuitively, one tunnel is destroyed and another one is created. As a consequence, objects
of Fig. 1a and Fig. 1b have the same topology. Notice that this kind of configuration has been considered, in particular in [10].
x
(a) (b)
Figure 1. A lump, made of 11 voxels, is depicted in (a). It contains no simple voxel, and is simple-equivalent to the complex in (b), made of 10 voxels. Both objects have three tunnels.
In this paper, we prove that in a particular case, a 3-D point can be removed while preserving topology if and only if it is a simple point. This property holds in the case of simply connected objects, that is, connected objects which have no tunnels.
We develop this work in the framework of abstract complexes. In this framework, we retrieve the main no- tions and results of digital topology, such as the notion of simple point.
2. Cubical complexes
Intuitively, a cubical complex may be thought of as a set of elements having various dimensions (e.g. cubes, squares, edges, vertices) glued together according to certain rules. In this section, we recall briefly some basic definitions on complexes, see also [4,3] for more details.
LetZbe the set of integers. We consider the families of
setsF10,F11, such thatF10={{a} |a∈Z},F11={{a,a+ 1} |a∈Z}. A subset f of Zn (n≥1) which is the Cartesian product of exactly m elements ofF11and(n− m)elements ofF10is called a face or an m-face ofZn, m is the dimension of f , and we write dim(f) =m.
We denote byFnthe set composed of all m-faces ofZn(m=0 to n). An m-face ofZnis called a point if m=0, a (unit) interval if m=1, a (unit) square if m=2, a (unit) cube if m=3. In the sequel, we will focus onF3.
Let f be a m-face inF3, with m∈ {0, . . . ,3}. We set ˆf ={g∈F3|g⊆f}, we say that ˆf is a cell or an m-cell. Any g∈ f is a face of f , and any gˆ ∈ f suchˆ that g6=f is a proper face of f .
A finite set X of faces ofF3is a complex (inF3) if for any f ∈X , we have ˆf ⊆X . Any subset Y of a complex X which is also a complex is a subcomplex of X . If Y is a subcomplex of X , we write Y X . If X is a complex inF3, we also write XF3. In Fig. 2 and Fig. 3, some complexes are represented.
Let XF3, a face f∈X is a facet of X if there is no g∈X such that f is a proper face of g. We denote by X+the set composed of all facets of X . The dimension of a non-empty complex X inF3is defined by dim(X) = max{dim(f)|f∈X+}. We say that X is an m-complex if dim(X) =m.
Let X F3 be a complex. A sequence π =
(f0, . . . ,fk) of 0-faces of X is a path in X (from f0 to
fk) if fi−1∪fi is a 1-face of X , for all i∈ {1, . . . ,k}.
The points f0and fkare the extremities of the path; the path is said to be a loop if f0= fk. The inverse of π is the path π−1= (g0, . . . ,gk) where gi= fk−i, for all i∈ {0, . . . ,k}. If π= (f0, . . . ,fk) andπ′= (h0, . . . ,hℓ) are two paths such that h0= fk, the concatenation of π andπ′ is the pathππ′= (f0, . . . ,fk,h1, . . . ,hℓ). If k=0, i.e.π= (f0), the pathπis called a trivial loop.
We say that X is connected if, for any two points f,g in X , there is a path in X from f to g. We say that Y is a connected component of X if YX , Y is connected and if Y is maximal for these two properties (i.e., we have Z=Y whenever YZX and Z is connected).
3. Topological invariants
Euler characteristics. Let X be a complex inF3, and let us denote by nithe number of i-faces of X , i= 0, . . . ,3. The Euler characteristic of X , writtenχ(X), is defined byχ(X) =n0−n1+n2−n3. The Euler charac- teristic is a well-known topological invariant. If X and Y are two complexes, we have the following basic prop- erty:χ(X∪Y) =χ(X) +χ(Y)−χ(X∩Y).
Fundamental group. The fundamental group, in- troduced by Poincar´e, is another topological invariant which describes the structure of tunnels in an object.
It is based on the notion of homotopy of loops. Briefly and informally, consider the relation between loops in a complex X , which links two loopsπandπ′wheneverπ can be “continuously deformed” (in X ) intoπ′(we say thatπ andπ′ are homotopic in X ). This relation is an equivalence relation, the equivalence classes of which form a group with the operation derived from the con- catenation of loops: it is the fundamental group of X .
Let us now define precisely the fundamental group in the framework of cubical complexes (see [7] for a similar construction in the framework of digital topol- ogy). Let X be a complex inF3, and let p be any point in X (called base point). Let Λp(X)be the set of all loops in X from p to p. Letπ,π′∈Λp(X), we say that π andπ′ are directly homotopic (in X ) if they are of the formπ=π1γπ2 andπ′=π1γ′π2, withγ and γ′ having the same extremities and being contained in a same face of X . We say thatπandπ′are homotopic (in X ), and we writeπ∼Xπ′, if there exists a sequence hπ0, . . . ,πℓisuch thatπ0=π,πℓ=π′, andπi,πi−1are directly homotopic in X , for all i∈ {1, . . . , ℓ}. The rela- tion∼X is an equivalence relation overΛp(X). Let us denote byΠp(X)the set of all equivalence classes for this relation. The concatenation of loops is compatible with the homotopy relation, i.e.,π1π2∼Xπ3π4when- everπ1∼Xπ3andπ2∼Xπ4. Hence, it induces an op- eration onΠp(X)which, to the equivalence classes of π1,π2∈Λp(X), associates the equivalence class ofπ1 π2. This operation (also denoted by) providesΠp(X) with a group structure, that is, (Πp(X),)satisfies the four following properties: closure (for all P,Q inΠp(X), PQ∈Πp(X)), associativity (for all P,Q,R inΠp(X), P(QR) = (PQ)R), identity (there exists an iden- tity element I∈Πp(X) such that for all P in Πp(X), PI=IP=P), and inverse (for all P inΠp(X), there exists an element P−1∈Πp(X), called the inverse of P, such that PP−1=P−1P=I). The identity element is the equivalence class of the trivial loop(p), and the inverse of the equivalence class of a loop π∈Λp(X) is the equivalence class of the inverse loopπ−1. The group(Πp(X),)is called the fundamental group of X with base point p.
If X is connected, it can be shown that the funda- mental groups of X with different base points are iso- morphic, thus in the sequel we will not refer anymore to the base point unless necessary.
We say that a group is trivial if it is reduced to the identity element. It may be easily seen that the funda- mental group of any single cell is trivial. A complex X 2
is said to be simply connected whenever it is connected and its fundamental group is trivial. Informally, sim- ply connected objects are connected objects which do not have tunnels. Such objects may have cavities (like a hollow sphere).
Let X be an n-complex, with n>0, and let m be an integer such that 0≤m≤n. We define the m-skeleton of X , denoted by Sm(X), as the subcomplex of X composed of all the k-faces of X , for all k≤m. The following property may be easily verified.
Proposition 1. Let XFn, letπ andπ′be two loops in X having the same base point. The loopsπ andπ′ are homotopic in X if and only if they are homotopic in S2(X).
Thus, the fundamental group of a complex XFn, for any n≥2, only depends on the 0-, 1- and 2-faces of X . The faces of higher dimension play no role in its construction.
4. Topology preserving operations
Collapse. The collapse, a well-known operation in algebraic topology [6], leads to a notion of homotopy equivalence in discrete spaces. To put it briefly, the col- lapse operation preserves topology.
Let X be a complex inF3and let f∈X+. If there exists one proper face g of f such that f is the only face of X which contains g, then we say that the pair(f,g) is a free pair for X . If (f,g) is a free pair for X , the complex Y=X\ {f,g}is an elementary collapse of X . In this case, we write XցeY .
Let X,Y be two complexes. We say that X col- lapses onto Y if X=Y or if there exists a collapse sequence from X to Y , i.e., a sequence of complexes hX0, . . . ,Xℓisuch that X0=X , Xℓ=Y , and Xi−1ցeXi, for all i∈ {1, . . . , ℓ}. Fig. 2 illustrates a collapse sequence.
We say that X and Y are collapse-equivalent if X=Y or if there exists a sequence of complexeshX0, . . . ,Xℓi such that X0=X , Xℓ=Y , and for any i∈ {1, . . . , ℓ}, ei- ther XiցeXi−1or Xi−1ցeXi holds. Let X,Y such that Y X . Obviously, if X collapses onto Y then X and Y are collapse-equivalent, but the converse is not true in general (a classical counter-example is Bing’s house, see [5,11]).
It is well known that, if two complexes X and Y are collapse-equivalent, then they have the same Euler characteristics and they have isomorphic fundamental groups.
Simplicity. Intuitively, a cell in a complex X is called simple if it can be “removed” from X while pre- serving topology. We recall here a definition of simplic-
f
(a) (b)
(c) (d)
(e) (f)
Figure 2. (a): a complex X , and a 3-face f . (f): a complex Y which is the detachment of ˆf from X . (a-f): a collapse sequence from X to Y .
(a) (b)
Figure 3. (a): the attachment of ˆf for X (see Fig. 2a). (b): the attachment of ˆx for the complex depicted in Fig. 1a.
ity (see [2]) based on the collapse operation, which can be seen as a discrete counterpart of the one given by T.Y. Kong [9].
The operation of detachment allows us to remove a subcomplex from a complex, while guaranteeing that the result is still a complex (see Fig. 2a,f). Let YX F3. We set X ⊘Y =∪{fˆ| f∈X\Y}. The set X ⊘Y is a complex which is the detachment of Y from X . Definition 2. Let XF3. Let f ∈X+, we say that f and ˆf are simple for X if X collapses onto X ⊘ f .ˆ
The notion of attachment leads to a local character- ization [2] of simple facets, which follows easily from the definitions. Let YXF3. The attachment of Y for X is the complex defined by Att(Y,X) =Y∩(X ⊘Y).
Proposition 3. Let XF3, let f∈X+. The facet f is simple for X if and only if ˆf collapses onto Att(ˆf,X).
In Fig. 2, we can check from the very definition of a simple face, that the 3-face f is indeed simple. As an illustration of Prop. 3, we can verify that the 3-face x of the complex depicted in Fig. 1a cannot collapse onto
its attachment, shown in Fig. 3; thus x is not simple.
5. The new property
In the image processing literature, a digital image is often considered as a set of pixels in 2-D or voxels in 3-D. A voxel is an elementary cube, thus an easy correspondance can be made between this classical view and the framework of cubical complexes. In the sequel of the paper, we call voxel any 3-cell. If a complex X F3is a union of voxels, we write X⊑F3. If X,Y⊑F3 and Y X , then we write Y ⊑X . From now on, we consider only complexes that are unions of voxels.
Notice that, if X⊑F3and if ˆf is a voxel of X , then X ⊘ fˆ⊑F3. There is indeed an equivalence between the operation on complexes that consists of removing (by detachment) a simple voxel, and the removal of a 26-simple voxel in the framework of digital topology (see [8,3]).
Let us quote a characterization of 3-D simple vox- els proposed by Kong in [9], which is equivalent to the following theorem for subcomplexes ofF3; this charac- terization will be used in the proof of our main theorem.
Theorem 4 (Adapted from Kong [9]). Let X⊑F3. Let f ∈X+. Then ˆf is a simple voxel for X if and only if Att(fˆ,X)is connected andχ(Att(ˆf,X)) =1.
Definition 5. Let X,Y ⊑F3. We say that X and Y are simple-equivalent if X=Y or if there exists a sequence of complexeshX0, . . . ,Xℓisuch that X0=X , Xℓ=Y , and for any i∈ {1, . . . , ℓ}, we have either
Xi=Xi−1 ⊘xi, where xi is a voxel that is simple for Xi−1; or
Xi−1=Xi ⊘xi, where xiis a voxel that is simple for Xi. We say that X is contractible if X is simple-equivalent to a single voxel.
Remark that, if X and Y are simple-equivalent then they are collapse-equivalent; hence they have the same Euler characteristics and their fundamental groups are isomorphic. We can now define the notion of lump evoked in the introduction.
Definition 6. Let Y⊑X⊑F3, such that X and Y are simple-equivalent. If X6=Y and X does not contain any simple voxel, then we say that X is a lump relative to Y , or simply a lump.
The following proposition will be used for the proof of Th. 8.
Proposition 7. Let X⊑F3 be a connected complex, let x be a voxel of X such that X ⊘x is connected and Att(x,X)is not connected. Then there exists a loop in X that is not homotopic in X to a trivial loop.
In other words, under the conditions of Prop. 7 the
complex X has a tunnel, more precisely it is not simply connected. A proof is given in the appendix, which fol- lows the same main lines as the proof of Prop. 3 in [1].
Finally, let us state and prove our main result.
Theorem 8. Let X⊑F3, such that X is simply con- nected. Let x be a voxel of X . Then x is simple for X if and only if X ⊘x is simple-equivalent to X .
Proof. The forward implication is obvious, let us prove the converse.
Suppose that X ⊘x is simple-equivalent to X and x is not simple for X . Remark that, since X is simply con- nected and X ⊘x is simple-equivalent to X , X ⊘x is also simply connected (for collapse preserves the fundamen- tal group). From the very definition of the attachment, we have χ(X) = χ([X ⊘x]∪x) = χ(X ⊘x) +χ(x)− χ(Att(x,X)). Since X and X ⊘x are simple-equivalent we haveχ(X) =χ(X ⊘x), furthermoreχ(x) =1 since x is a cell. From this we deduceχ(Att(x,X)) =1, hence Att(x,X)is non-empty, and from Th. 4, Att(x,X)can- not be connected. From Prop. 7, there exists a loop in X that is not homotopic to a trivial loop, thus the fun- damental group of X is not trivial, a contradiction with the fact that X is simply connected.
Since any contractible set is obviously simply con- nected, we have the following corollary.
Corollary 9. Let X⊑F3, such that X is contractible.
Let x be a voxel of X . Then x is simple for X if and only if X ⊘x is simple-equivalent to X .
6. Conclusion
We proved a new property about the notion of 3- D simple point, which has been extensively studied for fourty years and proved useful in many applications.
The interest of this result is not only theoretical, since configurations of the same nature as the lump of Fig. 1a are likely to appear in practical image processing pro- cedures (see [11]).
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Appendix: proof of Prop. 7
Proof. Let C1 and C2 be two distinct components of Att(x,X). Remark that C1and C2are subcomplexes of X ⊘x. Since X ⊘x is connected, there must exist a path γ1in X ⊘x that links a point p1∈C1to a point p2∈C2. Letγ2be a path from p2to p1in x;γ=γ1γ2constitutes a loop in X . We have seen that, in order to define the fundamental group, the base point can be arbitrarily chosen; the choice of a loop having p1as its extremities may thus be made without loss of generality.
For any pathπ, let us define the number #(π,C1) of pairs of consecutive points ofπthat are of the type (u,v)with u in C1and v not in x, or inversely. Obviously
#(γ2,C1) =0, and sinceγ1lies in X ⊘x and connects C1 to C2, it can be seen that #(γ1,C1)must be odd, hence
#(γ,C1)must also be odd.
Let us consider a loop γ′ directly homotopic in X to γ. We will prove in the following that #(γ′,C1) is odd. By induction, this property will extend to any loop homotopic in X toγ. By definition, we haveγ= P1QP2andγ′=P1RP2, with Q and R having the same extremities and being contained in a same face f of X . Observe that, by Prop. 1, we may suppose that f is a 1- or a 2-face. If ˆf x or if ˆf∩C1=/0, then obviously #(γ′,C1) =#(γ,C1). Suppose now that ˆf 6x and ˆf∩C16=/0.
Without loss of generality, we can write Q and R in the form Q=Q1Q′1Q2Q′2 . . .QkQ′k, k>0, and R= R1R′1R2R′2. . .RℓR′ℓ,ℓ >0, with all subsequences Qiand
Ribeing composed by points inside C1, all subsequences Q′iand R′ibeing composed by points outside C1, and all these subsequences being non-empty except possibly Q1, R1, Q′k, and R′ℓ.
Since ˆf∩C16=/0we have ˆf∩x6=/0, and since ˆf6x and ˆf X , we have ˆf∩xAtt(x,X). Hence, since C1
is a connected component of Att(x,X), we must have fˆ∩xC1. From this, we deduce that in ˆf , all the points that are not in C1are outside x, thus all the points in the subsequences Q′i and R′iare outside x.
Thus, the pairs Q′1Q2, Q2Q′2,. . .,Q′k−1Qk each bring a contribution of one unit to #(γ,C1). We have indeed: #(γ,C1) =#(P1,C1) +δ(Q1) +2k−3+ δ(Q′k) +#(P2,C1), where δ(π) = 0 whenever the path π is empty, and δ(π) = 1 otherwise. Remark that δ(Q1) =1 iff the first point of Q is in C1, and δ(Q′k) =1 iff the last point of Q is not in C1. Remark also that if k=1, then necessarily δ(Q1) =δ(Q′k) = 1. By the same reasonning, we have #(γ′,C1) =
#(P1,C1) +δ(R1) +2ℓ−3+δ(R′ℓ) +#(P2,C1), further- moreδ(Q1) =δ(R1)andδ(Q′k) =δ(R′ℓ)because Q and R have the same extremities. Since #(γ,C1)is odd, we see that #(γ′,C1)is also odd.
Hence the result, since for any trivial loop π we have #(π,C1) =0.
